[Paper Review] A note on Renner monoids

Eddy Godelle|ArXiv.org|Apr 6, 2009

Algebraic structures and combinatorial models17 references3 citations

TL;DR

This paper provides a monoid presentation and a new length function for all Renner monoids, extending Coxeter group theory to algebraic monoids. The length function generalizes the Coxeter length and behaves well with respect to Bruhat decomposition, with key results showing its compatibility with Hecke algebras and geometric formulas involving double coset dimensions.

ABSTRACT

We provide every Renner monoids with a monoid presentation, and we introduce a length function which extends the Coxeter length function.

Motivation & Objective

• To develop a Coxeter-like theory for Renner monoids, which play a role in linear algebraic monoid theory analogous to Weyl groups in algebraic group theory.
• To address the lack of a systematic monoid presentation and length function theory for Renner monoids, despite their structural importance.
• To introduce a new length function that extends the Coxeter length and exhibits favorable geometric and combinatorial properties.
• To establish a connection between this length function and the Hecke algebra of the Renner monoid, generalizing results from the rook monoid case.
• To provide a geometric formula for the length function using dimensions of double cosets in the Bruhat decomposition.

Proposed method

• Construct a monoid presentation for any Renner monoid $ R(M) $ using generators $ S igcup ilde{ ext{L}} $, where $ S $ is the Coxeter generating set and $ ilde{ ext{L}} $ is the set of non-identity elements of a cross-section lattice of idempotents.
• Define relations: (COX1) and (COX2) for the Coxeter group part; (TYM1), (TYM2), and (TYM) for interactions between generators and idempotents.
• Define the length function $ \ell $ by setting $ \ell(s) = 1 $ for $ s \in S $, $ \ell(e) = 0 $ for $ e \in \tilde{\text{L}} $, and extend additively to words, then take the minimal representative length.
• Use the normal decomposition $ (w_1, e, w_2) $ of elements in $ R(M) $ to analyze the behavior of the length function under multiplication by simple reflections.
• Establish a geometric formula: $ \ell(w) = \dim(Bw_1 e B) - \dim(B e w_2 B) $, where $ (w_1, e, w_2) $ is the normal decomposition of $ w $.
• Prove that the length function satisfies a Bruhat-type decomposition rule: $ BsBwB $ equals $ BwB $, $ BswB $, or $ BswB \cup BwB $, depending on whether $ \ell(sw) = \ell(w) $, $ \ell(sw) = \ell(w)+1 $, or $ \ell(sw) = \ell(w)-1 $.

Experimental results

Research questions

• RQ1 Can a monoid presentation be systematically constructed for all Renner monoids, generalizing the known presentation for the rook monoid?
• RQ2 Does a length function exist on Renner monoids that extends the Coxeter length and behaves well under multiplication by simple reflections?
• RQ3 Is there a geometric interpretation of the length function in terms of double coset dimensions in the Bruhat decomposition?
• RQ4 How does the new length function relate to the classical Solomon length function and to the Hecke algebra structure?
• RQ5 What conditions ensure that $ \ell(sw) = \ell(w) $, $ \ell(sw) = \ell(w)+1 $, or $ \ell(sw) = \ell(w)-1 $, and how do these relate to the normal decomposition?

Key findings

• The Renner monoid $ R(M) $ admits a monoid presentation with generators $ S \cup \Lambda_{\circ} $ and relations (COX1), (COX2), (TYM1), (TYM2), and (TYM), generalizing the rook monoid case.
• The new length function $ \ell $ satisfies $ \ell(sw) = \ell(w) $, $ \ell(sw) = \ell(w)+1 $, or $ \ell(sw) = \ell(w)-1 $, depending on the normal decomposition of $ sw $, as formalized in Proposition 0.2.
• The length function is geometrically realizable: $ \ell(w) = \dim(Bw_1 e B) - \dim(B e w_2 B) $, where $ (w_1, e, w_2) $ is the normal decomposition of $ w $.
• The new length function agrees with the Solomon length function up to a constant shift in the decomposition, as shown by $ \ell(sw) - \ell(w) = \tilde{l}(sw) - \tilde{l}(w) $, where $ \tilde{l} $ is the Solomon length.
• For $ w $ in the unit group $ W $, the length function reduces to the standard Coxeter length: $ \ell(w) = \dim(BwB) - \dim(B) $.
• The length function satisfies $ \ell(we) \leq \ell(w) $ and $ \ell(ew) \leq \ell(w) $, with equality if and only if the normal decomposition of $ we $ or $ ew $ is preserved in a specific way, as shown in Corollary 3.4.